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This is basically an Eulerian path problem dressed up in biology clothing, and I almost missed that.
Model the problem as finding an Eulerian path in a multigraph where tags are vertices and fragments are edges. Use Hierholzer's algorithm to find the path, then concatenate payloads in traversal order. Discuss handling disconnected components and edge cases like multiple valid paths.
Pro tip: Clarify whether the graph is guaranteed to have an Eulerian path; if not, discuss how to detect and handle cases with no solution. Also, mention that the payload concatenation order depends on the direction of traversal, so you must track edge orientation.
Restate the problem: given undirected fragments with two tags and a payload, find an ordering that uses each fragment exactly once and concatenate payloads. Confirm that tags can be reused and that fragments are undirected.
Treat each unique tag as a vertex and each fragment as an undirected edge connecting its two tags. The payload is associated with the edge. The goal is to find a trail that uses every edge exactly once (Eulerian trail).
For an undirected graph to have an Eulerian trail, all vertices with nonzero degree must belong to a single connected component, and the number of vertices with odd degree must be 0 or 2. If not, no solution exists.
Use Hierholzer's algorithm: start from a vertex with odd degree (if any) or any vertex with edges, perform DFS, and splice cycles. This yields the order of edges (fragments) in the trail.
Traverse the edges in the order found, and for each edge, append its payload to the result string. Ensure the direction of traversal matches the edge orientation (i.e., if you traverse from tag A to tag B, the payload is added as is; if reversed, still add the payload but note that the fragment is undirected so payload order doesn't change).
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